What’s Your Favorite Theorem?

A while ago I discovered a Podcast called My Favorite Theorem and have eagerly listened to each new episode. The basic format is that a mathematician is invited by the two hosts (Kevin Knudson and Evelyn Lamb) to describe his/her favorite theorem and also to pair the theorem with some non-mathematical thing, usually food or music.

What’s your favorite theorem?  Mine is the Chinese Remainder Theorem.  It’s got an obvious pairing, but that isn’t why I picked it.  My reason is that is appears in several courses I’ve taught and also has connections with my dissertation research way back in the 1970’s.

In an abstract algebra course, the Chinese Remainder Theorem says that if two positive integers, m and n, are relatively prime, then the ring of integers mod m \cdot n is isomorphic to the direct product the rings of integers mod m and n.

In number theory, the same fact is framed differently, that the system of congruences x \equiv a \pmod{m} and x \equiv b \pmod{n} always has a unique solution mod m\cdot n as long as m and n are relatively prime.

My dissertation research involved approximation and interpolation of functions, and when you generalize the Chinese Remainder Theorem to Euclidean Domains, one immediate implication is that given n+1 points on the plane (pick any field) with distinct x-values, there is always a unique polynomial of degree n or less that passes through the points.

If you have a favorite theorem feel free to post it in the comments!

UML Participation in the 2019 William Lowell Putnam Mathematics Competition

UML Students working on the Putnam.

What a way to spend your Saturday! Get yourself to campus for 10 AM and work on six math problems for three hours. Then after a two hour break, spend another three hours of six more problems. That’s what thousands of undergraduate students throughout the US and Canada, including 34 UMass Lowell students, did on December 7 to take part in the 2019 William Lowell Putnam Mathematics Competition.

The competition, sponsored by the Mathematical Association of America, took place concurrently throughout the US and Canada. Last year,  4,623 students from 568 institutions participated. There were two 3 hour sessions, each with six problems. As usual, the problems were tough. Here is probably the easiest of them:

Determine all possible values of the expression
A3 +B3 +C3 – 3 A B C,
where A, B, and C are nonnegative integers.

A complete list of problems: 2019 Putnam Problems

Professor Kenneth Levasseur served as supervised competition at UML.   Thanks to the Honors College for providing refreshments for the students on the day of the event.

Results will be announced in late March.